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ARITHMETIC THE SECOND
sive class of primes p. He did not succeed in proving Fermat's
theorem for all primes; certain slippery Exceptional primes'
eluded Kummer's net - and still do. Nevertheless the step ahead
which he took so far surpassed everything that all his prede-
cessors had done that Kummer became famous almost in spite
of himself. He was awarded a prize for which he had not
competed.
The report in full of the French Academy of Sciences on the
competitition for its 'Grand Prize' in 1857 ran as follows.
'Report on the competition for the grand prize in mathematical
sciences. Already set in the competition for 1853 and prorogued
to 1856. The committee, having found no work which seemed to
it worthy of the prize among those submitted to it in competi-
tion s proposed to the Academy to award it to M. Kummer,
for his beautiful researches on complex numbers composed of
roots of unity* and integers. The Academy adopted this
proposal.'
Kummer's earliest work on Fermat's Last Theorem is dated
October 1835. This was followed by further papers in 1814 3-7,
the last of which was entitled Proof of Fermafs Theorem on the
Impossibility of yP -j- yp = zpfor an Infinite^ Number of Primes
p. He continued to add improvements to his theory, including
its application to the laws of higher reciprocity, till 1874, when
he was sixty-four years old.
Although these highly abstract researches were the field of
his greatest interest, and although he said of himself, *To
* If ZP 4- y& = #', then #P = SP — yt>t and resolving SP —t/*> into its
p factors of the first degree, we get
SUP « (z-y) (z-ry) (z-r*y) ... (a-i*-^),
in •which r is a *pth root of unity' (other than 1), namely i* — 1 = 0,
with r not equal to 1. The algebraic integers in the field of degree p
generated by r are those which Kummer introduced into the study of
Ferfnafs equation, and which, led him to the invention of his *ideal
numbers' to restore unique factorization in the field - an integer in
such a field is not uniquely the product of primes in the field for a8
primes p.
t The 'infinite* in Kummer's title is stul (1936) unjustified; 'many*
should be put for 'infinite*.
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